Draw line after path
I did something similar in the past, although I was looking for rounded bezier paths and I came up with this, which works - somehow - also when lines are straight, and not rounded. I'm not sure if it helps you but at least it's done differently, as it does not use bevel but an animated array modifier plus a curve modifier...

this is the setup: the bezier segments iirc have been straightened using: V => vector first, and then V => free.

I also exaggerated the line width (editing the small rectangle in edit mode) to better show it in pictures here.



and this is the animation that is produced:



I'm not sure if it solves your corners issues...

Here's the file, anyway.